THE DEFINABLE CONTENT OF HOMOLOGICAL INVARIANTS I: Ext & lim1

The paper’s Theorem 5.11 proves a definable isomorphism lim^1 A ≅ Â/A for any filtration A of a countable abelian group A, constructing a continuous map σ: Z(A) → Â by σ(a) = lim_k a_{0,k} and showing it induces a definable isomorphism on the quotients (Â/A) (; set-up and definitions appear in and the completion functor in ). The candidate solution defines the same forward map in inverse-limit coordinates, σ(x)_m = x_{0,m} mod A(m), and a Borel inverse τ using countable Borel sections. These maps are inverse modulo B(A), yielding the same definable isomorphism. The arguments differ in presentation (Borel sections vs. telescoping limits), but they implement the same identification and rely on the same identities in Z(A).